514 Proceedings of Royal Society of Edinburgh. [july 18, 
Here there are three symmetric systems of quantities 
(«!•») ? (^1‘m) 5 (m V n) ) 
the first appearing in every column of equations, the second in 
every row, and the third only once. The determinants of these 
systems are denoted by 
D m , S n , M„ , 
respectively : that is to say 
T) M = S(:h ^1*1 ^2*2 • • • ®»'n) 
& n = S( ± a ri a 2 . 2 . . . a n . n ) 
M n = S(±m 1 . 1 w 2 . 2 . . . m. n . n ) . 
If now in 
+ * * • ®n*») 
there he substituted for m rl , m 1>2 their values as given by 
the group of equations, there will be obtained a function of all the 
a’s and as, which must be an alternating function with respect to 
the first indices of the cC s and also with respect to the first indices 
of the a’s. Further, since each of the m's is of the first degree in 
the a * 8 and of the first degree also in the a’s, each term of the 
development of S(±m 1 . 1 m 2 . 2 .... m n . n ) must evidently be of 
the form 
— ®'1’M®'2" I/ • • • • • . . • Cl n . n . 
But the development by reason of its double alternating character 
cannot contain such a term without containing all the terms of the 
product 
ih d~ oq*ju, a - 2 , v • ' * • • • a n‘Tt 5) • 
Consequently it must equal one or more products of this kind. But 
again the indices fi, v, . . ., ir are either all different or not. If 
they be different, we have 
S(±a rjtA a 2 . v . . . a n . ff ) = ± S(± a rl a 2 . 2 . . . a„. n )=±8„; 
and if any two of them be equal 
it a i’M- a 2 ’V * * * 7r) = ^ * 
The like is true in regard to S ± (a rix ,a 2 . v , . . . a n . n ,). This enables 
us to conclude that the development of M n is equal to one or more 
products of the form 
± T> n 8 n : 
M n = cDJ n , 
in other words, that 
